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Mengoni, M and Ponthot, JP (2015) A generic anisotropic continuum damage model integration scheme adaptable to both ductile damage and biological damage-like situations. International Journal of Plasticity, 66. pp. 46-70. ISSN 0749-6419

https://doi.org/10.1016/j.ijplas.2014.04.005

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A generic anisotropic continuum damage model

integration scheme adaptable to both ductile damage

and biological damage-like situations

M. Mengonia,∗, J.P. Ponthota

aUniversity of Liege (ULg), Department of Aerospace and Mechanical Engineering,Non-linear Computational Mechanics (MN2L), Liege, Belgium

Abstract

This paper aims at presenting a general versatile time integration scheme appli-

cable to anisotropic damage coupled to elastoplasticity, considering any damage

rate and isotropic hardening formulations. For this purpose a staggered time

integration scheme in a finite strain framework is presented, together with an an-

alytical consistent tangent operator. The only restrictive hypothesis is to work

with an undamaged isotropic material, assumed here to follow a J2 plasticity

model. The only anisotropy considered is thus a damage-induced anisotropy.

The possibility to couple any damage rate law with the present algorithm is il-

lustrated with a classical ductile damage model for aluminium, and a biological

damage-like application. The later proposes an original bone remodelling law

coupled to trabecular bone plasticity for the simulation of orthodontic tooth

movements. All the developments have been considered in the framework of the

implicit non-linear finite element code Metafor (developed at the LTAS/MN2L,

University of Liege, Belgium - www.metafor.ltas.ulg.ac.be).

Keywords: Continuum Damage Mechanics, anisotropic material,

elastic-plastic material, constitutive behaviour, numerical algorithms

∗Corresponding author.Email address: mmengoni@ulg.ac.be (M. Mengoni)

Preprint submitted to International Journal of Plasticity February 17, 2014

1. Introduction

Damage mechanics deals with modelling the loss of stiffness and progressive mi-

croscopic failure mechanisms induced by external loading in a material. Coupled

damage models are models in which damage is incorporated into constitutive

equations. Their use can lead to the development of complex constitutive equa-

tions whose numerical integration has to be considered adequately.

Two main coupled approaches to damage mechanics can be found in the litera-

ture.

The first one is a micromechanical approach to damage based on the work

of Gurson (1977) who considered the growth of spherical voids in a plastic

material. The extensions to the Gurson-Tvergaard-Needlemann (GTN) model

accounted for plastic hardening and physically described the ductility of mate-

rials (Tvergaard and Needleman, 1984; Rousselier, 1987). GTN models define

the damage variable as the void fraction, and its evolution is due to the nucle-

ation, growth and coalescence of voids. While initially an isotropic approach to

damage, it has been extended to anisotropic damage (Hammi and Horstemeyer,

2007; Zapara et al., 2012; Horstemeyer and Bammann, 2010).

The second one is a phenomenological approach to damage often referred to as

the Continuum Damage Mechanics (CDM). It should be noted however that

the GTN approach is also a continuum approach to damage and thus that the

terminology CDM may be considered as improper. Phenomenological dam-

age models based on the concept of effective stress space were introduced by

Kachanov (1958) and later by Rabotnov (1968) (as cited in Voyiadjis and Kat-

tan, 2006; Lemaitre and Desmorat, 2005) who were the first to introduce a

scalar damage variable which may be interpreted as the effective surface density

of micro-cracks per unit volume. The CDM is derived from a thermodynamic

framework, ensuring the positivity of the dissipation. The damage variable is an

internal variable related to the effective density of cracks or cavities at each point

(for the isotropic case) or at each point and in each direction (anisotropic case),

that is, to the microstructure. As for the GTN approach, while the CDM was

2

initially developed considering a scalar damage variable, it has been extended

to a tensor representation describing a damage-induced anisotropy (Voyiadjis

et al., 2008; Desmorat and Otin, 2008; Brunig et al., 2008; Abu Al-Rub and

Voyiadjis, 2003; Badreddine et al., 2010; Brodland et al., 2006; Dunand et al.,

2012).

For both approaches, a description of the coupling between damage and elasto-

plasticity has to be completed with a damage evolution law. While damage

evolution laws were proposed in the original works deriving the GTN or CDM

approaches, several other damage rate expressions can be found in the literature

describing different damage mechanisms (Duddu and Waisman, 2013; Souza and

Allen, 2012; Tekoglu and Pardoen, 2010; Hammi and Horstemeyer, 2007; Qi and

Bertram, 1999; Kitzig and Haußler-Combe, 2011; Lecarme et al., 2011; Khan

and Liu, 2012; Lai et al., 2009; Zaıri et al., 2011) and coupling with time-, rate-,

and temperature-dependent materials (Abu Al Rub and Darabi, 2012; Besson,

2009; Stewart et al., 2011; Horstemeyer et al., 2000; Guo et al., 2013), to cite

only a few.

In the present work, we will use the phenomenological approach of anisotropic

damage, i.e. anisotropic Continuum Damage Mechanics. This choice is driven

by the wide range of applications of CDM. In particular, and as will be treated

further in this work, it can be extended to represent a stiffness softening not

linked to the growth of micro-cracks but to other phenomena, such as a bio-

chemical coupling. Considering biological effects coupled to mechanical loading,

we will be interested in describing the effect of biological actions on the material

behaviour rather than describing in details the biological phenomena. The use

of a CDM approach in this context rather than a GTN approach of damage

is thus straightforward to describe the evolution of the stiffness tensor due to

external loading. Besides, the coupling between elasticity and damage in those

cases plays a strong role which is naturally included into the CDM approach. In

the case of GTN models, the elastic properties can also be functions of damage

but this dependence is often neglected.

In this work an additive decomposition of the strain rate is assumed to model

3

the elastoplasticity in finite strains. It should however be noted that several

studies also developed mathematical frameworks to couple anisotropic damage

and multiplicative elastoplasticity (Menzel et al., 2005; Brunig, 2002, 2003a;

Ekh et al., 2004). This approach to elastoplasticity in large strains leads to

a completely different formulation of damage (Brunig, 2003b). The computa-

tional tools developed to integrate anisotropic damage in that case are thus not

applicable in the present work.

The numerical integration algorithms of constitutive models incorporating anisotropic

damage effects presented in the literature (Lemaitre and Desmorat, 2005; Borgqvist

andWallin, 2013; El khaoulani and Bouchard, 2013; Brunig et al., 2008; de Souza Neto

et al., 2011) are usually limited to one given damage model and are not eas-

ily extended to other formulations or damage criteria. The numerical scheme

in Simo and Ju (1987a,b) or in Jeunechamps and Ponthot (2013) is similar to

the proposed approach in such a way that the integration can be considered

as a triple operator split: elastic predictor, plastic corrector, damage corrector.

However this numerical scheme was developed for isotropic damage, i.e. with a

scalar equation to solve for the damage correction and direct decoupling of dam-

age and plasticity for the plastic correction. For those two reasons, several other

generic integration schemes have been proposed in the case of isotropic dam-

age (de Souza Neto et al., 1994; de Souza Neto and Peric, 1996; Doghri, 1995;

Vaz and Owen, 2001; Mashayekhi et al., 2005; Boers et al., 2005). Simo and

Ju (1987a,b) also propose a numerical scheme for the integration of anisotropic

damage. However, in that case, a strain-based anisotropic damage is assumed,

thus leading to a different numerical approach.

The present work is aimed at developing a generic phenomenological anisotropic

damage integration scheme that can be coupled with any isotropic hardening

law and damage rate. The only restrictive hypothesis was to work with a ma-

terial whose undamaged behaviour can be modelled with a von-Mises elasto-

plastic model. The anisotropy of the material is thus only a damage-induced

anisotropy. Otherwise, any thermodynamically consistent damage criterion can

be used in conjunction with the proposed original staggered integration scheme.

4

Furthermore a finite strains assumption is considered for the anisotropic con-

tinuum damage formulation thus allowing the use of the model in large strains

and large rotations applications. The developed algorithm can be also coupled

to damage criteria as different as criteria describing ductile damage or biologi-

cal damage-like phenomena. Considering the later case, an original, enhanced

extension of a small-strain elastic-damage model is developed in this work and

applied to simulate a tooth displacement in an orthodontic treatment. This

work thus presents an original fully coupled non-linear model of bone remod-

elling occurring during orthodontic tooth movement.

Beyond this introduction, this paper is divided into three main sections. The

first section presents the extension of CDM to a finite strain formulation con-

sidering an anisotropic symmetric second order damage tensor. Using a second

order tensor restricts the anisotropy to orthotropy. This extension makes no as-

sumption on the damage rate except that it remains a symmetric tensor. A new

implicit time integration algorithm in a finite element context is then proposed.

The following two sections demonstrate the versatility of the approach. First a

ductile damage model was used to verify the proposed approach by comparison

to another integration scheme from the literature (de Souza Neto et al., 2011).

For this, a simple uniaxial test was reproduced to compare the present results

with those of Aboudi (2011). Second, a biologically driven damage model was

developed. Its aim was to propose a model of orthodontic tooth movement

considering biological softening and hardening of bone tissue.

Notations

Einstein summation convention is used except when indicated.

As a general rule, scalar a, σ is denoted by a light-face italic letter; second-

order tensor a, σ, D is signified by boldface italic letter; fourth-order tensor

A is identified by blackboard character. Compact tensor notation is used as

much as possible. The double dot product, or double contraction, is signified

with a : b = aijbij and [A : b]ij = Aijklbkl. The dyadic product A = a ⊗ b

and the symmetrized outer product B = a⊗b are defined as Aijkl = aijbkl and

5

Bijkl = 1/2(aikbjl + ajlbik), respectively. Letter I signifies the second-order

identity tensor with the components being Kronecker-delta δij , and 1 is the

symmetric fourth-order unit deviatoric tensor: 1 = I⊗I − 1/3I ⊗ I. |D| isa tensor whose principal components are the absolute value of the principal

components of D: |D| =3∑

i=1

|di|~ni ⊗ ~ni (no Einstein summation) with di and

~ni the eigenvalues and eigenvectors of D.

2. A staggered integration procedure for anisotropic continuum dam-

age theory

The principles of Continuum Damage Mechanics (CDM) introduce a fictitious

undamaged configuration (called the effective configuration). In this configura-

tion the (scalar) damage is virtually removed in such a way that applying the

actual loads leads to an effective stress, σ, defined as (Lemaitre and Desmorat,

2005):

σ =σ

1− d(1)

where d is a direction independent damage variable, and σ is the Cauchy stress

tensor.

2.1. Anisotropic Continuum Damage Mechanics

To ensure a more general formulation of the principles of damage mechanics, the

case of anisotropic damage is assumed. In this case different levels of damage

are related to the principal directions of the physical space, and thus a simple

scalar damage parameter is no longer sufficient to quantify damage in all direc-

tions. Instead, the anisotropy of the damage distribution in the material is here

interpreted using a symmetric second-order damage tensor, d.

Effective stress

The effective stress tensor in the case of anisotropic damage is defined in such

a way that it is a function of the second order damage variable and of the

6

stress tensor. To keep a linear relationship between the effective and actual

configurations, one often writes the effective stress so that

σ = M(d) : σ (2)

The fourth order tensor M is not a damage tensor, it simply is function of a

second order damage tensor d, the internal variable which is normalized and

related to the current state of the microstructure. Several definitions of M can

be found in the literature (Abu Al Rub and Voyiadjis, 2006; Lemaitre et al.,

2000; Lemaitre and Desmorat, 2005; Lennon and Prendergast, 2004; Menzel

et al., 2002; Voyiadjis and Kattan, 2006), we will here adopt the one proposed

by Lemaitre et al. (2000):

M = H⊗H − 1

3

(H2 ⊗ I + I ⊗H2

)+

1

9tr(H2)I ⊗ I +

1

3

I ⊗ I

1− ηdH(3)

where dH = tr (d) /3 is the hydrostatic damage, η is an hydrostatic sensitivity

parameter, and H is the effective damage tensor defined as:

d = I −H−2 (4)

This fourth order tensor, M, has the properties of being

• symmetric, therefore resulting in a symmetric effective stress (using a sim-

ple extension of the isotropic definition: Mijkl = δik[I−d]−1jl does not lead

to a symmetric result);

• independent of the Poisson’s ratio value;

• compatible with the thermodynamics in a small strain approach: existence

of a state potential from which the constitutive law can be derived and of

a principle of strain equivalence (the symmetrization M = I⊗(I −d)−1 is

not derived from a potential);

• able to represent different effects of damage on the hydrostatic and devi-

atoric stresses by means of an hydrostatic sensitivity parameter, η.

7

Combining equ.(2) and equ.(3) gives an equivalent stress expressed as a function

of the deviatoric part of the Cauchy stress s and pressure p by:

pI =p

1− ηdHI = p M : I

s = dev (HsH) = M : s(5)

The definition of the effective stress equ.(2) can be analytically inverted as:

σ = M−1 : σ

with M−1 = H−1⊗H−1 − H−2 ⊗H−2

tr(H−2

) +1

3

(1− ηdH

)I ⊗ I

leading to

pI =(1− ηdH

)pI = p M−1 : I

s = H−1sH−1 − s : H−2

tr(H−2

)H−2 = M−1 : s(6)

Strain equivalence formulation of damage

The strain equivalence approach of damage relates the stress level in the dam-

aged material with the stress in the undamaged material that leads to the same

strain. This assumes the deformation behaviour is affected by damage only

through the effective stress. In small strain elasticity, Hooke’s law is then writ-

ten:

σ = Ho : ε

where Ho is Hooke’s fourth order tensor of elasticity, with parameters evaluated

for the undamaged material:

Ho = KI ⊗ I + 2G1

with K and G respectively the bulk and shear moduli of the undamaged mate-

rial.

2.2. Elasto-plasticity in finite strains

Assuming a constitutive law for the undamaged material which is not restricted

to basic small strain linear elasticity, we work under a finite strain framework

8

with an additive decomposition of the strain rate:

D = De +Dp (7)

The next assumption is to work in a corotational frame. From this point forward,

all quantities are expressed in a corotational frame. In this paper, we use no

distinct notation for corotational quantities. All the time derivatives noted • are

thus to be understood as derivatives in the corotational frame. These derivatives

can be associated to objective derivatives in a fixed frame of reference. In the

present case, it is equivalent to a Jaumann rate (Ponthot, 2002). Thus, in the

corotational frame, we assume that the corotational effective stress rate is linked

to the elastic part of the corotational strain rate by the generalized Hooke’s law

in an hypoelastic formulation, in such a way that:

•

σ = Ho : De (8)

When coupling damage and plasticity, one needs to express the yield criterion

as well as the hardening laws in terms of damaged variables (details can be

found in Lemaitre and Desmorat (2005); Desmorat and Otin (2008) for the

small strains version or in Jeunechamps and Ponthot (2013) for an extension

to finite strains of isotropic damage in a hypoelastic framework). Using a von-

Mises criterion for the undamaged material leads to an equivalent stress defined

in the effective stress configuration as:

σvMeq =

√

3

2s : s (9)

The yield function then becomes:

f = σvMeq − σy ≤ 0 (10)

where σy is the (current) yield stress.

Assuming a constant damage tensor (this assumption is consistent with the

staggered scheme of integration that will be described later, see section 2.3) and

a normality rule in the Cauchy stress space (it is the space where the conser-

vation equations are fulfilled and the Clausius-Duhem inequality is respected)

9

gives the following flow rule (details are given in Appendix A):

Dp = ΛN (11)

where Λ is the consistency or flow parameter and with the deviatoric unit nor-

mal, N , defined by:

N =

∂f

∂σ||∂f/∂σ|| =

n

||n|| (12)

with the notation

n = M : s = dev (HsH) (13)

The equivalent plastic strain rate is then given by:

•

εp

=

√

2

3Dp : Dp =

√

2

3Λ (14)

The damaged equivalent plastic strain, εp,d, is the internal parameter driving

the plastic problem. This new parameter is conjugated to the yield limit of the

material, σy, and defined through its rate,•

εp,d

, as:

Dp =•

εp,d ∂f

∂σ⇒

•

εp,d

=Λ

||∂f/∂σ|| =√

2

3Λ||s||||n||

Finally, the last assumption for the plastic problem is to restrict the hardening

law to be isotropic:

σy(εp,d) = σ0

y + hεp,d

with h =dσydεp,d

the hardening parameter and where σ0y is the yield limit of the

virgin material.

This plastic problem is coupled to the damage problem, defined through the

damage evolution law. In this general theoretical background, we assume that

the damage rate can be described with any general damage model and we simply

write:•

H =•

H(σ,D, εp,d,•

εp,d

). It will be particularized to two specific rates

further.

2.3. Time integration algorithm

The equations that are driving the coupled damage elasto-plastic problem are

the definition of the yield criterion, i.e. one scalar equation, and the decompo-

10

sition of the strain rate and the damage rate, i.e. two tensorial equations:

Strain rate decomposition D = De +Dp

Yield function f = σvMeq − σy(

•

εp,d

, εp,d) ≤ 0

Effective damage tensor evolution•

H =•

H(σ,D, εp,d,•

εp,d

)

(15)

This system introduces one scalar and two tensorial unknowns, i.e. the damaged

equivalent plastic strain rate•

εp,d

, the elastic strain rate De, and the damage

evolution•

H. Accounting for the symmetry of De and H, this is therefore a

non-linear system of 13 scalar unknowns.

This principal system is completed with the definition of the effective stress, the

constitutive model, and the plastic flow rule, i.e. 3 tensorial equations, and the

definition of the damaged equivalent plastic strain rate, i.e. a scalar equation:

Definition of the effective stress σ = M : σ

Constitutive law•

σ = Ho : De

Plastic flow rule Dp = Λdev (HsH)

||dev (HsH) ||Damaged equivalent plastic strain rate

•

εp,d

=

√

3

2Λ||s||||n||

(16)

This introduces 19 new secondary unknowns: the Cauchy stress tensor σ (6

unknowns), the effective stress tensor σ (6 unknowns), the plastic strain rate

Dp (6 unknowns), and the flow parameter Λ (1 unknown).

Solving the system given by equation (15) with a fully coupled Newton-Raphson

method would require to evaluate the derivatives for each of the 13 variables

with respect to each other variables, i.e. 169 derivatives. The computational

cost of such a method is therefore high and considered as an issue to solve such a

problem. Thus, the principle of the integration scheme proposed for the present

coupled problem (plasticity and anisotropic damage) is a staggered scheme. As

mentioned earlier, using such a scheme allows us to derive the driving equations

of the plastic problem independently from the damage evolution law. It also

allows us to derive a generic resolution scheme that can be coupled to any

damage rate. The complete scheme of integration, inspired from Jeunechamps

11

and Ponthot (2013) who proposed an integration scheme for isotropic continuum

damage coupled to von-Mises plasticity, can be described as follows (see Fig. 1):

• Starting from known values at time n in the corotational frame (σn,Hn, εp,dn )

the plastic problem is solved at constant damage in the effective stress

space (the procedure to solve the plastic problem is detailed hereafter).

So we first determine the effective stress, σn+1, and the damaged equiva-

lent plastic strain, εp,dn+1.

• From this result, the damage evolution (through the effective damage

tensor variation) is computed at constant stress and plastic strain to

obtain the final value of the effective damage tensor Hn+1. The new

value of the effective damage tensor is computed using an explicit scheme

on the time derivative over the time-step: Hn+1 = Hn +•

H∆t with•

H =•

H(σn+1,Hn, εp,dn+1, ...) evaluated with the new effective stress and

damaged equivalent plastic strain rate but with the effective damage ten-

sor evaluated at the end of the previous time step: Hn.

• This solving procedure is done iteratively, with iterations on the effective

damage tensor ending when the norm of the difference between two con-

secutive tensors is below a user-chosen tolerance (TOLd, chosen by default

as 10−7).

This stress integration problem is solved in the corotational effective stress space.

The corotational Cauchy stress is then obtained simply by applying the inverse

of the anisotropic damage operator on the effective stress (equ.(6)).

The detailed algorithm used to solve the plastic problem (box "Solve Plastic

Problem" in Fig. 1) is detailed hereafter. Using the additive decomposition of

the strain rate, the constitutive law (equ.(8)) becomes:

•

σ = Ho : D −H

o : Dp (17)

As seen in equ.(11), (12) and (13), the plastic strain rate is purely deviatoric

and the trace of the elastic strain rate tensor is equal to the one of the total

strain rate tensor.

12

σn , dn , εp,dn known

Initiate loop on damage

k= 0 ; H(k)n+1 = Hn = ( I-d n ) 1/2

Elastic predictor

pn+1 = pe = pn + K tr( )∆EN

se = sn + 2G ∆E

N

Solve Plastic Problem(constant damage tensor)

to get s(k)n+1 , ε

p,d (k)n+1

Update effective damage tensor

(constant stress tensor and plastic strain rate)

H(k+1)n+1 = Hn +

•H(s(k)

n+1, pn+1, H(k)n+1, ε

p,d (k)n+1

, ...) ∆t

Convergence?

||H (k+1)n+1 - H

(k)n+1 || ≤TOLd

k= k+ 1

YES

NO

Compute damage tensor

dn+1 = I - H(k+1)n+1

2

Compute Corotational stress

σn+1 = -1(dn+1 ) :σn+1

Figure 1: Outline of the integration scheme for the coupled problem (elasto-plasticity and

anisotropic damage) in the corotational frame, σ, d, εp,d are known at time step n and

need to be computed for time step n+ 1. ∆EN = tr

(

∆EN)

I +∆EN

is the natural strain

increment over the time step, see Ponthot (2002)

13

Considering an isotropic behaviour for the undamaged material, the hydrostatic

and deviatoric parts of the effective stress rate therefore become:

•

p = Ktr (D)•

s = 2Gdev (D)− 2GΛN(18)

with K the undamaged material bulk modulus and G its undamaged shear

modulus.

The time integration is realised thanks to an elastic predictor/plastic corrector

algorithm (closest point algorithm)1 in a time-step procedure where the stress

tensor at time n+1 is computed from the stress tensor at time n in an iterative

setup (Newton-Raphson algorithm).

The elastic predictor accounts only for the elastic strain rate and gives:

pe = pn +Ktr(

∆EN)

se = sn + 2G∆EN

(19)

with ∆EN

= dev (lnU) and tr(

∆EN)

= tr (lnU) for U the right stretch

tensor evaluated between the configurations at time n and time n + 1 (incre-

mental strain resulting from the polar decomposition F = RU over the time

step).

If needed (i.e. if f = σvMeq − σy > 0), a plastic correction is performed for the

deviatoric stress so that:

sn+1 = se − 2GΓNn+1

= se − 2GΓM : sn+1

||M : sn+1||(20)

with Γ =

∫ tn+1

tn

Λ dt and with the normal to the yield surface N computed at

n+ 1 as a closest point projection scheme is used.

The internal variable driving the plastic problem is updated by:

εp,dn+1 = εp,dn +

√

2

3Γ

||sn+1||||M : sn+1||

(21)

1During the elastic predictor step, D = De and D

p = 0 while for the plastic correction (if

needed, i.e. if the elastic predictor does not satisfy the yield criterion) D = 0 and Dp = ΛN

14

where Γ is computed through a Newton-Raphson scheme to fulfil the yield

criterion.

This procedure (equ.(20) and (21)) for computing sn+1 and Γ however results

in a fully coupled non-linear system. The resolution of this system through a

classical Newton-Raphson method unfortunately results in strong convergence

problems. To overcome this, we can modify this system so that we get a linear

system for the deviatoric stress integration. We can indeed choose to write the

plastic strain rate in terms of a normal which is not a unit normal, such as n

(equ.(13)) instead of N (equ.(11)):

Dp = λn (22)

We then have

λ =Λ

||n|| =Λ

||M : s|| (23)

Thus, the plastic correction (equ.(20)) becomes:

sn+1 = se − 2GγM : sn+1 (24)

with γ =

∫ tn+1

tn

λ dt the scalar plastic increment.

The plastic correction can thus be written in the form of a linear system in

sn+1:

P : sn+1 = se (25)

where P = I⊗I + 2GγM.

The internal variable driving the plastic problem is then computed as:

εp,dn+1 = εp,dn +

√

2

3γ||sn+1|| (26)

To solve the plasticity problem, γ has to be computed so that f = σvMeq (s) −

σy(εp,d) = 0.

Starting with γ0 as the value of the plastic multiplier at the end of the previous

15

time-step, γi at the ith iteration of the Newton-Raphson algorithm is given by:

γi+1 = γi +∆γ

ri = σvMeq (γi)− σy(γ

i)

∆γ = − ri

∂σvMeq

∂γ

∣∣∣∣∣γ=γi

− ∂σy∂γ

∣∣∣∣γ=γi

(27)

The value of γ is iterated on untilri+1

σy(γi+1)< TOLγ with TOLγ a user-chosen

tolerance (chosen by default as 10−7).

Details on the way to solve the deviatoric stress (equ.(25)) and to compute the

yield criterion derivative with respect to γ (equ.(27)) are given in Appendix B.

The coupled problem (plasticity/anisotropic damage) has to be completed by an

evolution law for the damage tensor. A priori, any thermodynamically consis-

tent damage evolution could be used with this anisotropic damage elasto-plastic

model. Compared to other integration algorithms used for anisotropic damage

coupled to elastoplasticity (Lemaitre and Desmorat, 2005; Borgqvist and Wallin,

2013; El khaoulani and Bouchard, 2013; Brunig et al., 2008; Menzel and Stein-

mann, 2001; Abu Al-Rub and Voyiadjis, 2003), this staggered procedure has

the advantage that it makes no assumption on the damage evolution other than

keeping a symmetric damage tensor. It can thus be used for different materials

having different damage mechanisms. It also leads to the derivation of a generic

closed-form expression of the consistent tangent operator that can be applied to

any damage evolution or isotropic hardening models provided they fit into the

expression given in equ.(15). The derivation of this operator is detailed in Ap-

pendix C. It is however restricted to a material whose undamaged behaviour

can be modelled with a von-Mises elasto-plastic model and isotropic harden-

ing. The same principle of integration can be used with other plastic criteria if

the plastic update can be written as the canonical expression equ.(25). Kine-

matic hardening can be considered if the corresponding terms of the consistent

tangent operator are adapted. Finally, this staggered algorithm decouples the

fully coupled non-linear system with 13 scalar unknowns into two smaller quasi

16

linear systems: one of 7 scalar unknowns for the plasticity problem, nonlinear

in only the plastic multiplier, and one of 6 scalar unknowns, the anisotropic

linear damage problem. On a computational point of view, it is thus obvious

that the decoupled system requires less operations at each iteration for its res-

olution than the fully coupled one. Considering isotropic damage, it has been

shown (Jeunechamps and Ponthot, 2013) that the staggered algorithm is indeed

more efficient with respect to the CPU cost.

Two examples of completely different background are detailed in the next sec-

tions.

All numerical developments described have been implemented into the non-

linear implicit object-oriented finite element code Metafor (developed at the

LTAS/MN2L, University of Liege, Belgium - www.metafor.ltas.ulg.ac.be)

used in this work.

3. Application to ductile damage as a verification process

The modelisation of damage processes in ductile failure has been widely cov-

ered in the literature (Bonora et al., 2006; Brunig and Gerke, 2011; Luo et al.,

2012; Mahnken, 2002; Shojaei et al., 2013). Recent reviews of models of duc-

tile failure can be found in Besson (2010); Horstemeyer and Bammann (2010);

Jeunechamps and Ponthot (2013) and references therein. Using a phenomeno-

logical approach of damage rather than a more physically motivated model has

been regularly discussed and is judged controversial (Besson, 2009). The aim

of this section however is neither to propose a new model of ductile failure nor

to simulate an actual process but rather to demonstrate the versatility of the

integration process presented in this work and to verify its implementation. For

this purpose, the results of the simple uniaxial test of a ductile metallic ma-

trix of a composite material proposed in Aboudi (2011) are reproduced. This

matrix is made of a 2024-T4 aluminium alloy and its mechanical behaviour

at constant temperature can be modelled using the approach presented in this

work. All material parameters needed to model the elasto-plastic behaviour are

17

Table 1: Material parameters of the 2024-T4 aluminium alloy at 21oC (Aboudi, 2011)

Young’s Poisson’s Yield Hardening

modulus ratio stress coefficient

(GPa) (−) (MPa) (GPa)

72.4 0.33 286.7 11.7

described in Table 1. In his work, Aboudi uses the anisotropic damage evo-

lution model of Lemaitre et al. (2000) as the aluminium alloy used is one of

the material against which Lemaitre et al.’s model has been validated in the

case of uncoupled plasticity. This model is a direct extension of the original

isotropic Lemaitre’s model (Lemaitre, 1992) and assumes the damage evolution

is led by the plastic strain rate. It was initially built and used in an infinitesimal

strain framework and is here extended to a finite strain formalism. The damage

tensor evolution derived from a damage dissipation potential is written as:

•

d =

(

Y

S

)s

|Dp| if εp ≥ εth (28)

where S, s and εth are material parameters, the latter allowing for the damage

to exist only above a threshold given in terms of the accumulated plastic strain

and Y is the effective elastic strain energy expressed in terms of the triaxiality

function as:

Y =σvM 2

eq

2E

(

2

3(1 + ν) + 3(1− 2ν)

(p

σvMeq

)2)

︸ ︷︷ ︸

triaxiality function

(29)

The principal directions of the damage rate are thus aligned with those of the

plastic strain rate.

All material parameters needed to model the anisotropic damage are described

in Table 2.

The problem studied in Aboudi (2011) and reproduced here is a uniaxial load-

ing/unloading cycle up to 2% natural strain of a single hexahedral finite ele-

ment. In this study a 8 nodes selectively reduced integration element (constant

volumetric strain) is used. Results of the damage eigenvalues evolution and

18

Table 2: Damage parameters for the 2024-T4 aluminium alloy (Aboudi, 2011)

S s εth η

(MPa) (−) (−) (−)

0.05 1 0. 2.6

the Cauchy stress are represented in Fig. 2 and 3 and compared to the results

in Aboudi (2011). Fig. 2 shows the damage tensor rate induces anisotropy.

Indeed, the damage eigenvalue d1 increases twice as much as the two other

eigenvalues, proportionally to the plastic strain eigenvalues. The stiffness in the

corresponding directions is inversely proportional to the damage eigenvalues,

thus producing a transversely isotropic stiffness. A difference below 0.5% is ob-

served between the results obtained with the present approach and the results

of Aboudi (2011). This difference can be explained by the way the numerical

algorithm to integrate the constitutive law is considered converged. Indeed, the

approach of de Souza Neto et al. (2011) used by Aboudi (2011) evaluates the

convergence over the whole coupled system including plastic strain increment

and damage tensor, d. However, in the present approach, the constitutive law is

considered as converged when the increment of the effective damage tensor, H,

is stationary. Fig. 3 shows the effect damage has on the material. The mate-

rial hardening is reduced by comparison to the undamaged state in the loading

case. The unloading has a lower slope than the loading one as the damage effect

on the elastic stiffness has a softening effect. Finally the damage effect on the

plastic unloading is a softening effect while the undamaged material still follows

a hardening law.

Fig. 4 shows the effect the hydrostatic parameter η has on the damage evolution.

The end value of the damage eigenvalues decreases within increasing η. This

effect of η depends however on the triaxiality as it weights the hydrostatic and

deviatoric part of the damage rate. Here as only a uniaxial test is performed,

the triaxiality function is constant. For a variable triaxiality, the effect of η is

not however as simple as observed in the present case.

19

0 0.5 1 1.5 20

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

d , proposed formulation1

d , Aboudi (2011)1

d , d , proposed formulation2 3

d , d , Aboudi (2011)2 3

(log strain)11 (%)

damage principal component (-)

Figure 2: Damage eigenvalues vs. natural strain for a loading-unloading uniaxial test.

0 0.5 1 1.5 2

-400

-300

-200

-100

0

100

200

300

400

600

-600

proposed formulationAboudi (2011)

s11 (MPa)

no damage

(log strain)11 (%)

Figure 3: Cauchy stress vs. natural strain for a loading-unloading uniaxial test.

20

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

2.5

11.52

damage principal component d (-)1

(log strain) (%)11

h

Figure 4: Damage eigenvalue vs. natural strain for a loading-unloading uniaxial test and

different values of η.

This simple example showed the damage integration approach presented in this

work produces similar results when compared to more computationally expen-

sive coupled approaches such as the integration scheme of de Souza Neto et al.

(2011). It also showed a simple extension of the original Lemaitre damage rate

to anisotropy is sufficient to imply large orthotropic effects in the material.

Finally, the effect of damage on the elastic part of the deformation is clearly

highlighted in a loading/unloading cycle as the unloading elastic slope is lower

than the loading one.

4. Damage-like biological application: A model for orthodontic tooth

movement

This section aims at proposing a biomechanical application of the anisotropic

damage framework presented. This framework has already been applied to

bone (Zysset and Curnier, 1996; Garcia et al., 2009), articular cartilage (Hos-

seini et al., 2014), intervertebral disc (Qasim et al., 2014), and other soft tis-

sues (Bernick et al., 2011; Balzani et al., 2012; Calvo et al., 2009). In this

21

A

A’

AA’

trabecular bone

cortical bone

Figure 5: Alveolar bone of a mandibular left canine (79-year-old male). Trabecular (or can-

cellous) internal structure and cortical shell (stl file from Boryor et al. (2009)).

work, an orthodontic tooth movement model is developed and applied to two

representative types of tooth movements.

4.1. Generic principles of orthodontics and bone remodelling

The basic principle in orthodontics is to gradually impose progressive and ir-

reversible bone deformations using specific force systems on the teeth. In or-

thodontic treatments, the tooth displacement is mainly due to bone remodelling

that leads the teeth into new positions (Krishnan and Davidovitch, 2009; Masella

and Chung, 2008; Lindauer, 2001; Melsen, 2001). Both cortical an cancellous

bone (see Fig. 5), as adapted, adjustable, and optimized living structures, are

constantly renewed. This permanent remodelling is an adaptive process that

beyond biological aspects depends on the mechanical loading on the skeleton.

It aims at preserving the mechanical properties of the bone and adapting its

structure in response to the mechanical demands it experiences. It decreases

the amount of bone where it is of no mechanical relevance; while it increases

its density to reinforce bone where it is necessary. Besides the density change,

remodelling also occurs to modify the bone topology, mainly in trabecular bone

tissue for which the trabeculae tend to align along the principal stress directions.

22

Bone remodelling therefore depends not only on the load intensity but also on

the load directions.

From a phenomenological point of view and as explained by the mechanostat

theory (Frost, 1964, 1987), remodelling occurs in order to homogenize at the

tissue level a stimulus, ψt, in the neighbourhood of an homeostatic tissue level

value, ψ⋆t . This stimulus is a scalar representation of the applied mechanical

loads. Remodelling can be modelled by relating the bone density rate,•

ρ, to a

remodelling rate,•

r (Beaupre et al., 1990).

•

ρ = kSvρ0•

r (30)

with ρ the apparent density and ρ0 the density of the fully mineralised bone

tissue. The terms kSv in equ.(30) take into account the available (k ∈ [0, 1])

bone specific surface area (Sv internal surface area per unit volume). They thus

express the fact that a bone surface has to exist for bone cells to act and induce

remodelling. k accounts for the fact that all this surface is not available for

the cells to act. The specific surface area Sv can be related to the porosity

f = 1− ρρ0.

The remodelling rate•

r is a function of the difference between the current value

of the chosen stimulus, ψt, and its homeostatic counterpart, ψ⋆t . For anisotropic

models, the change in trabecular orientation is modelled through a change of

the stiffness directionality related to the external loads direction. Most models

also assume the existence of a “dead” or lazy zone (an interval, of half-width ω,

around the homeostatic level for which no remodelling process takes place). The

remodelling rate needs to be defined is such a way that remodelling takes place

to resorb bone (•

r < 0) and decrease the apparent density where “underloaded”

conditions are encountered, ψt < ψ⋆t −ω, while formation of bone (

•

r > 0) occurs

(increase of density) where “overloaded” conditions are encountered, ψt > ψ⋆t +

ω. These conditions for remodelling are usually called the remodelling criteria

and are sketched on Fig. 6.

The original model which is proposed in this work is built on a damage/repair

based approach of remodelling. It is a phenomenological model, stated first by

23

2w

ytyt*

r

c f

cr

lazy

zoneunderload overload

Figure 6: Remodelling rate as a function of the remodelling stimulus.

Doblare and co-workers (Doblare and Garcıa, 2002; Garcıa et al., 2002). This

model has been chosen as a framework because it is one of the few models whose

stimulus variation is justified through thermodynamical concepts of Continuum

Mechanics. It is also one of the few models for which the remodelling law is

fully integrated into the constitutive model. In the case of bone remodelling,

“damage” has to be understood as a measure of the bone topology (bone density

and trabecular orientation). The measure of damage used is therefore virtual

and actually reflects the bone density and orientation that can evolve. There is

no actual damage in the tissue. The undamaged material is the ideal situation

of bone with null porosity and perfect isotropy, i.e. the material considered

for the fully mineralised bone. The process of bone resorption corresponds

to the classical damage evolution concept, since it increases the void fraction

and therefore damage. However, bone apposition can reduce damage and lead

to bone repair. Damage repair can be considered because the total energy

dissipation also includes biological dissipation due to metabolism on top of the

mechanical dissipation which is negative for damage repair (Jacobs, 1994).

In the present work, Doblare and Garcıa (2002); Garcıa et al. (2002)’s model

is firstly enhanced to be coupled to an elasto-plastic material behaviour in a

finite strain framework (see Mengoni and Ponthot (2010) for an isotropic version

of this elasto-plastic damage coupling). The new model can therefore capture

24

permanent strains of the tissue beyond the ones due to the density variation only.

The bone tissue topology is here considered as an anisotropic “organization” of

elasto-plastic trabeculae (although it is clear that the relevant inelastic process

is different from that of the classical metal plasticity). This “organization” is

measured through morphological parameters: a mean bone density, or bone

volume fraction i.e. bone volume over total volume: BV/TV or ρ = ρ/ρ0, and

a fabric tensor (T as defined in Cowin (1986)) that measures of the amount

of bone tissue in a given direction and describes the anisotropy. The plastic

behaviour may not be relevant in bone remodelling applications due to the very

low strain levels involved, permanent strains due to density variations are much

higher than the one due to a plastic behaviour. However, the proposed model

can be used both in low strain levels and higher strain levels problems keeping

the continuum representation of the topology through the use of the fabric tensor

and allowing to represent a plastic behaviour of the bone trabeculae.

The remodelling model is secondly slightly modified to account for complex

biological behaviour of the periodontal ligament, i.e. the soft tissue membrane

laying between the tooth and its supporting alveolar bone (see section 4.3).

Even tough it was not the approach followed here, it should be noted that Voyi-

adjis and Kattan (2006) proposed generic anisotropic damage models related to

the morphology through the use of fabric tensors. They used fourth order dam-

age tensors, related to a fabric tensor representing crack distributions. Their

aim was to provide a sound physical meaning to an anisotropic representation

of damage in elastic theories.

4.2. Phenomenological remodelling model expressed in the Continuum Damage

Theory Framework

To apply Doblare and Garcıa (2002); Garcıa et al. (2002)’s model within the

theoretical framework presented earlier, the effective stress definition is rep-

resented in a strain equivalence approach of damage. The yield criterion is

expressed for the undamaged material, here the fully mineralised bone. Actu-

ally, this anisotropic model is applied to trabecular bone only while the cortical

25

bone is considered as initially isotropic and remaining isotropic during remod-

elling, thus following an isotropic version of this anisotropic model (Mengoni

and Ponthot, 2010).

We thus define an anisotropic damage tensor that depends on morphological

parameters (BV/TV or ρ and fabric tensor: T ) by the expression:

d = I − ρβAT (31)

whereA is a calibration functions that allows to retrieve the classical formulation

of damage in the isotropic case, and β is the exponent of the power law relating

ρ to the bone Young’s modulus, E (β can be function of ρ), i.e. E ∝ ρβ (Jacobs,

1994).

This definition of damage fulfils the requirements of a damage variable, i.e.

d = 0 for ρ = 1 and T = I (pure isotropy), corresponding to the undamaged

state, and d = I for ρ = 0 and any value of T ; which means complete absence

of bone mass. Using a normalization condition for T , such as tr (T ) = 3, yields

to the independence of the two internal variables, d and ρ (Doblare and Garcıa,

2002).

Considering equ.(4) and (31), we see that the effective damage tensor H has its

principal directions aligned with the fabric tensor T principal directions:

H−2 = ρβAT

This effective damage tensor includes not only the directionality of the bone

microstructure through the fabric tensor (T ), but also the porosity by means of

the density (ρ). A remodelling law affecting both ρ and T will thus be translated

into a damage evolution law.

In order to derive an evolution law for the effective damage tensor H , or equiv-

alently for the fabric tensor T , we define an external mechanical stimulus, Y ,

identified as the variable thermodynamically conjugated to the effective damage

tensor in terms of the free energy density function (Ψ).

Y = −∂Ψ(σ,H)

∂H(32)

26

This free energy density is calculated considering an isotropic material behaviour

at trabecular level (and assuming we can extend the expression of the free energy

density in small strains to a finite strain problem) and expressing it in terms of

Cauchy stress as:

Ψ =1

2

(p2

K(1− ηdH)+

1

2Gtr (HsHs)

)

(33)

Y is then obtained in terms of the external independent variable (Cauchy stress,

p, s) and the internal variable (effective damage tensor,H) as (deriving equ.(33)

with respect to H):

Y =1

3K

ηp2

(1− ηdH)2H−3 +

1

2GsHs (34)

The damage criterion is the domain of the external mechanical stimulus, Y , for

which damage is not modified (the lazy zone as used in the literature of bone

remodelling) both in overload and underload conditions. Following Doblare and

Garcıa (2002); Garcıa et al. (2002)’s approach and adapting it to the present

formalism, we propose two damage criteria, go and gu, one for overloaded con-

ditions and one for underloaded ones:

Overload: go = C31/4√1− w

(J : J)1/4 − (ψ⋆t + ω)ρ2−β/2 (35)

Underload: gu = −C 31/4√1− w

(J : J)1/4 + (ψ⋆t − ω)ρ2−β/2 (36)

with J = W : Y =1

3(1− 2w)tr (Y ) I +wY .

Considering an associated evolution law for the effective damage tensor, we can

write:•

H = µo ∂go∂Y

+ µu ∂gu∂Y

(37)

where µo and µu are flow parameters with the consistency conditions µo, µu ≥0; go, gu ≤ 0; µogo = µugu = 0.

Deriving the remodelling criteria (equ.(35) and (36)), combining with the ef-

fective damage tensor definition (equ.(4)), and the density variation (equ.(30)),

the evolution law of the effective damage tensor becomes for both underload

27

and overload conditions:

•

H = −βkSv•

r

2

tr(H−2

)

tr(H−3(W : J)

)ρ0ρW : J (38)

As two parameters, w in the definition of W, and η in the definition of σ, are

defined to weigh the deviatoric and hydrostatic parts of tensors, and as they

are defined on two distinct intervals (w ∈ [0, 1[ and η ∈ [1,+∞[), we actually

use w = 1 − e−(η−1) to reduce the number of parameters to one. This (w,η)

mapping is a continuous mapping between a parameter, η, defined in an infinite

interval and one, w, defined in a finite interval. As η is usually bounded (about

3 for metals (Lemaitre and Desmorat, 2005) and taken to be equal to the degree

of anisotropy in the case of bone), the choice of this relationship induces no

numerical difficulties (while w would tend to 1 for large values of η and the

division by (1− w) in equ.(35), (36) would not be possible otherwise).

The remodelling rate•

r is obtained from the remodelling criterion that is cur-

rently active (see Fig. 6):

•

r =

−crgu

ρ2−β/2for gu ≥ 0, go < 0,

0 for gu < 0, go < 0,

cfgo

ρ2−β/2for go ≥ 0, gu < 0,

(39)

The parameters cr and cf are related to the remodelling velocity. They are

usually chosen are constant such as in Fig. 6. They can however generally be

function of the remodelling stimulus or another measure of stress.

Finally, the density variation can be computed from equ.(30). It has to be

emphasized however that this density change is not actually used to compute

a change of mass. Only the stiffness variation due the density rate is used as

the density is considered having an influence on stiffness only. Therefore, there

is no actual change of mass considered in the bone. The terms formation and

resorption are thus employed as increase and decrease of stiffness respectively.

28

4.3. Application to Orthodontic Tooth Movement

Such a remodelling model cannot be used straight away in an orthodontic tooth

movement problem. Indeed in this case, the remodelling is driven not only by the

bone cells reactions to the applied loads but also by the periodontal ligament’s

cells (Krishnan and Davidovitch, 2009; Masella and Chung, 2008; Lindauer,

2001). The periodontal ligament is the soft membrane laying between the bone

and the teeth. Its role is not only to be a buffer membrane between two hard

tissues but also to provide a blood flow to these tissues (Melsen, 2001; Verna

et al., 2004), blood flow which is necessary for the bone cells to induce remod-

elling. When forces are applied to the tooth by orthodontic devices, the blood

flow within the ligament is disrupted and so is the cell activity. This change of

cell activity (mainly the fibroblasts) triggers remodelling, which is thus strongly

related to the hydrostatic pressure (Van Schepdael et al., 2013). In such a sit-

uation, bone resorption will be observed for underload situations and overload

situations in compression while bone formation will arise in overload situations

in tension only. In order to account for this difference in the model, a slight

change is made in the definition of the remodelling rate. Instead of using a

remodelling rate such as defined in equ.(39) with a constant remodelling coef-

ficient cf in overloaded situations, we define a piecewise constant remodelling

coefficient in overloaded situations that accounts for all phenomenon in the pe-

riodontal ligament. This coefficient c depends on the hydrostatic stress in such

a way that it is positive in tension, negative in compression, and null for high

compression situations where the blood and cell flow is completely interrupted

(Fig. 7).

This model (anisotropic remodelling adapted to orthodontic problem and ex-

pressed in a Continuum Damage framework) is here applied to a characteristic

situation in orthodontics, a tooth translation (bodily tooth movement in 2D

and intrusion in 3D) into its supporting bone (see Fig. 8). The bodily tooth

movement is modelled as an applied displacement in the case of a 2D analy-

sis. The intrusion is due to an applied force in a 3D analysis. In both cases,

the tooth is considered rigid, the periodontal ligament mechanical behaviour

29

p

c

-cr

cf

pcrittensioncompression

formation

resorption

Figure 7: Value of the remodelling coefficient c as a function of the hydrostatic pressure for

the remodelling law adapted to the specific case of orthodontic tooth movement in overloaded

conditions.

follows a non-linear interface model (Mengoni, 2012), and the bone follows me-

chanical and remodelling laws such as described in this work. Both trabecular

and cortical bone are initially isotropic. Trabecular bone can evolve to be-

come anisotropic while it is assumed that cortical bone remains isotropic. The

consideration of a von-Mises plasticity for the trabecular level has been dis-

cussed previously (Bayraktar and Keaveny, 2004; Mengoni et al., 2012; Verhulp

et al., 2008) and is considered here as a valid assumption. Such a criterion ho-

mogenised with a fabric tensor leads to an apparent level plastic criterion that

can be characterised as a general Hill criterion in the orthotropy axis defined

by the fabric tensor (Mengoni, 2012). All material and remodelling parameters

of the modelled tissues are described in Table 3. In particular, at tissue level,

the yield limit is chosen to be 200 MPa (median value of the one calculated

with a reverse engineering approach on linear finite element models in Niebur

et al. (2000); Stolken and Kinney (2003); Verhulp et al. (2008)). The isotropic

hardening parameter is computed so that the tangential modulus is set to 5%

of the value of Young’s modulus as in the previously cited studies. The remod-

elling parameters are not particularized to orthodontic models and are those of

Doblare and Garcıa (2002). However, given that for the initial state, for which

there is equilibrium, ψ = 0 and that only orthodontic loads are considered, the

reference stimulus, ψ⋆, is here taken at zero also. A more realistic model ac-

30

Table 3: Material and Remodelling parameters of bone tissues

Remodelling parameters

ψ⋆ cr cf ω η

(MPa) (µm/(dayMPa)) (µm/(dayMPa)) (MPa) (−)

0 10 5 0.1 1

Mechanical parameters

Material Young’s Poisson’s Yield Hardening Apparent

modulus ratio stress parameter density

(GPa) (−) (MPa) (MPa) (gr/cc)

Bone at tissue level 13.75 0.3 200.0 723.7

Trabecular bone 0.95

Cortical bone 1.805

counting for all forces and non-zero stimulus at equilibrium would allow a better

choice of reference stimulus. This choice of reference stimulus will in particular

not allow modelling the loss of bone following a tooth loss. It indeed does not

account for possible loss of stiffness due to underuse.

The tissues geometries were extracted from CT-scans images (obtained from the

OSIRIX image dataset2). These CT images of a mandible were segmented into

cortical bone, trabecular bone, and teeth (see Fig. 8). For the 2D analysis (plane

strain), a slice of the left central incisor in the mesiodistal plane was extracted

and a geometry using cubic splines for the tooth, trabecular bone, and cortical

bone was built. This geometry was then meshed with linear quadrilaterals with

constant pressure. For the 3D analysis, a multiple material surface mesh was

constructed (d’Otreppe et al., 2012), and a volumic mesh built using linear

tetrahedra.

2OSIRIX/INCISIX dataset http://www.osirix-viewer.com/datasets/

31

Lingual

side Labial

side

trabecular

bone

cortical

bone

tooth

Figure 8: Geometries and boundary conditions of the orthodontic tooth movement models.

Left: 2D incisive submitted to bodily tooth movement, the bottom dashed curve is constrained.

Right: 3D incisive submitted to an intrusion movement, the surface within the bottom box is

constrained.

32

Bodily tooth movement

The 2D problem is a displacement driven problem. The tooth is translated

perpendicularly to its main axis (see Fig. 8) over a distance corresponding to

the periodontal ligament width and kept at that position over a period of six

months. The effect of remodelling can be observed globally by a decrease of

the force needed to sustain such a displacement and locally by a density and

orientation change within the bone. Bone density variation is observed during

this constant displacement period, leading to a reduction of the needed force

as depicted in Fig. 9 (plain curve). The initial force intensity needed to move

the tooth is 1.24N, i.e. a value that can be compared to clinically applied

forces (Roberts, 2000; Bourauel et al., 2000). This initial force causes high

hydrostatic stress in the compression side, impeaching remodelling to occur as

c (Fig.(7)) is zero. Thus initially only bone formation can occur in the tension

side of the tooth (right in the figure), increasing the needed force to maintain

the displacement. On the compression side, peripheral (with respect to the

bone/tooth interface) remodelling slowly decreases the density (see Fig. 10, left)

and thus the compressive hydrostatic stress. The force needed to maintain the

displacement then decreases due to direct remodelling once the hydrostatic stress

at the compression side has been reduced (after about forty days see Fig. 10,

right). A force reduction by a factor of two is observed in the following 20 days.

After 150 days of remodelling, the force has been reduced by six and stabilizes at

that asymptotic level further on. We can also notice that after 70 days (Fig. 10,

right zoomed area), the contours of the bone at the bone/tooth interface matches

those of the tooth, as remodelling tends for the bone to follow the tooth. This

was obviously not yet the case after 35 days (Fig. 10, left zoomed area) where a

space between tooth and bone exists representing the stretching or compression

of the periodontal ligament due to mechanical loading. After a certain amount

of time, this altered situation of the periodontal ligament is eliminated thanks

to the bone remodelling process. If the same model is considered with a faster

remodelling (e.g. doubling the remodelling constants cf and cr), the behaviour

33

0 20 40 60 80 100 120 140 160 180

time (days)

0.4

0.8

1.2

force (N)

Figure 9: Displacement driven tooth movement - intensity of the force (N) needed to maintain

a displacement over time (days). Plain curve: translation movement, and dashed curve:

translation with a remodelling constant twice as high as the previous one.

is identical to the previously described but with a twice as fast response (Fig 9,

dashed curve). Those constants are indeed the only drivers of time in the model

of remodelling developed in this work. Fig. 10 also shows a ring shape area

(in blue) on the lower left side of the trabecular bone where high resorption

has occurred. The presence of this area is only a numerical boundary condition

artefact. As its effect does not extend to the base of the tooth root, it does not

perturb the remodelling around the tooth.

The remodelling also has an effect on the trabecular orientation in the bone.

This effect can be assessed by analyzing the stiffness eigenvalues and eigenvectors

that represent the principal directions. As seen in Figure 11, while the initial

configuration was an isotropic bone, after 150 days it is completely different.

The trabecular bone is highly oriented around the tooth to be stiffer in the

direction of displacement while it remained almost unaffected in the basal bone.

34

tooth and

bone line

merged

0.55 1.8

bone density gr/cc( )

35 days

tooth

line

bone

line

0.15 1.9

bone density gr/cc( )

70 days

Figure 10: Displacement driven tooth movement - bone density [gr/cc] after 35 days (left) and

after 70 days (right). Initial values for trabecular bone: 0.95 gr/cc, for cortical bone: 1.805

gr/cc

35

Figure 11: Trabecular bone orientation - Movement of translation after 180 days. The ellipses

represent the stiffness eigenvalue (the axis orientations are those of the fabric tensor eigen-

vectors and their lengths are proportional to the fabric tensor eigenvalues), a 1GPa isotropic

indicator is also depicted on the right side of the tooth.

36

Tooth intrusion

This 3D problem is a force driven problem. Due to applied forces, the tooth

is intruded into the bone (see Fig. 8). This movement is obtained applying a

0.7N force on the tip of the crown, towards the apex of the tooth (the force is

applied with a constant direction, whatever the tooth movement). The force is

kept constant for three months while the bone density variation observed during

this period leads to an increase of the tooth displacement.

The obtained displacement is directed towards the tooth apex, however due to

the nonlinearity of the periodontal ligament, it is composed of a translation

and a slight rotation (see Fig. 12). The initial displacement of 0.18mm (see

Fig. 13) is due to the periodontal ligament deformation mainly. It creates high

compression of the bone along most of the tooth/bone interface, thus impeaching

direct remodelling. The remodelling thus takes initially place either away from

the bone/tooth interface (peripheral resorption), or to form new bone. The

first remodelling effect is thus a very low decrease of the displacement. As seen

in Figure 13, the displacement rate increases after about 15 days when direct

remodelling is then possible. The tooth displacement (Fig. 13), while initially

decreasing because of the bone formation apically, slowly increases to reach an

increase of 50% of the initial (instantaneous) displacement.

5. Conclusions

This paper presented two main original contributions.

First, a numerical integration procedure for an anisotropic continuum damage

model coupled to elasto-plasticity was proposed in a finite strain context. It is a

staggered scheme of integration instead of a fully coupled integration procedure

often found in the literature. This allowed us to reduce the computational

cost of such an integration algorithm. Indeed the staggered scheme solves two

decoupled systems of 7 scalar unknowns (for the plasticity problem, nonlinear in

only the plastic multiplier) and 6 scalar unknowns (the anisotropic linear damage

problem) instead of a fully coupled nonlinear system with 13 scalar unknowns.

37

hydrostatic stress ( )MPa

-7.0 7.00.0

Figure 12: Intrusion: hydrostatic stress (MPa, positive in tension) after a month of remod-

elling. The blue arrows represent the application of the intrusion force. The brown lines

represent the displacement field, enhancing the rotation behaviour

38

0 10 20 300

0.05

0.1

0.15

0.2

0.25

displacementmm( )

timedays( )

40 50 60 70 80 90

0.3

Figure 13: Intrusion: displacement of the tooth (mm) vs. time (days)

This integration scheme was verified against a simple uniaxial test from the

literature. Moreover, the proposed algorithm allows exact linearisation, leading

to the derivation of a generic closed-form expression of the consistent tangent

operator. This quadratic rate of convergence of the full Newton-Raphson scheme

is thus preserved.

Second, a constitutive model able to simulate the coupled biological and mechan-

ical phenomenon within the bone in orthodontic tooth movement applications

was developed. Compared to other tooth movement models of the literature,

the remodelling behaviour is fully included into a non-linear constitutive law

for the bone. This constitutive model is built on morphological parameters to

describe the bone anisotropy, accounting for effects such as plasticity of the tra-

beculae, and for which the continuum parameters such as the stiffness can evolve

with morphology as remodelling occurs in the tissue. In spite of the necessary

idealizations, the proposed phenomenological description of bone remodelling

specified for alveolar bone allows to qualitatively represent density variation

of the bone surrounding a tooth when submitted to loading representative of

orthodontic appliances. Simple cases of orthodontic tooth movements can be

represented with the present model when applied to geometries reconstructed

from CT-scans images. However the presented applications did not simulate

39

the effect of actual orthodontic devices, in particular further work should focus

on a more realistic representation of boundary and initial conditions. Indeed,

we assumed that the only forces present during orthodontic tooth movement

where the orthodontic forces, neglecting all physiological forces (muscle tone,

chewing forces and other mouth activities, as well as gravitational forces). We

thus used a value of the remodelling stimulus that was not physiological but

was representative of the equilibrium state of the proposed computational ap-

proach. Moreover, it has been shown that remodelling algorithms such as the

one used here exhibit a high sensitivity to boundary conditions. Obtaining a

better computational representation of those conditions is thus a necessary step

to produce predictive orthodontic tooth movement models.

The approach taken here in the development of the integration scheme was

general and allowed for two completely different damage rates to be used (duc-

tile damage and biomechanical tissue adaptation). Its implementation into an

object-oriented finite element code facilitates the addition of any new damage

models coupled to any new isotropic hardening laws.

40

Appendix A. Deriving the Plastic Flow Rule for the Anisotropic

Damage Model

The plastic strain rate is assumed to follow the normality rule (associated plas-

ticity) in the Cauchy stress space. For a von-Mises yield function, it gives:

Dp = ΛN = Λ

∂f(s)

∂σ||∂f/∂σ||

with f = σvMeq − σy and

∂f(s)

∂σ=∂f(s)

∂s

∂s

∂σ

Dp = Λ

∂f(s)

∂σ||∂f/∂σ|| = Λ

s :∂s

∂σ

||s :∂s

∂σ||

(A.1)

Upon writing

s = dev (HsH) = HsH − 1

3tr (HsH) I (A.2)

one gets (assuming H constant3)

∂sij∂σkl

=∂HimsmnHnj

∂σkl− 1

3

∂HqrsrsHsq

∂σklδij

=

[

HikHlj −1

3HimHmjδkl

]

− 1

3

[

HqkHlq −1

3HqrHrqδkl

]

δij (A.3)

One can therefore write (s being deviatoric et H symmetric4)

sij∂sij∂σkl

= sij

[

HikHlj −1

3HimHmjδkl

]

= HkisijHjl −1

3HmisijHjmδkl (A.4)

or equivalently

s :∂s

∂σ= dev (HsH) (A.5)

and eventually

Dp = ΛN = Λdev (HsH)

||dev (HsH) || (A.6)

3This assumption is valid considering a staggered scheme of integration for the coupled

plastic/damage problem.4The symmetry of H has to be ensured by its initial symmetry and a symmetric evolution

function.

41

The flow direction (unit normal to the yield function) is given by

N =

∂f

∂σ||∂f/∂σ|| =

dev (HsH)

||dev (HsH) || (A.7)

This unit normal fulfils the following properties:

N : N = 1 N : dN = 0 (A.8)

The equivalent plastic strain rate is therefore given by:

•

εp

=

√

2

3Dp : Dp = Λ

√

2

3N : N =

√

2

3Λ (A.9)

We also define n as:

n = dev (HsH) = M : s (A.10)

Using A.7, this results in:

N =M : s

||M : s|| =n

||n|| (A.11)

Appendix B. Solving the Plastic Problem

The equations that drive the plastic update are as follows (equ. 24 and 26)

sn+1 = se − 2GγM : sn+1 (B.1)

εp,dn+1 = εp,dn +

√

2

3γ||sn+1|| (B.2)

and the update of the plastic multiplier in the Newton-Raphson process can be

written:

∆γ = − ri

∂σvMeq

∂γ

∣∣∣∣∣γ=γi

− ∂σy∂γ

∣∣∣∣γ=γi

(B.3)

The first equation (B.1) is a linear system in s. It can be written in the form:

P : sn+1 = se (B.4)

where

Pijkl =1

2(δikδjl + δilδjk) + 2GγMijkl = P(γ,H)

42

The solution to this system is:

sn+1 = S : se (B.5)

where S is the inverse of P. Notice however that P is not actually inverted. S is

used as a notation, a linear system has to be solved to compute sn+1.

The update of the plastic multiplier (B.3) requires to compute∂σvM n+1

eq

∂γ

∣∣∣∣∣γ=γi

and∂σn+1

y

∂γ

∣∣∣∣∣γ=γi

.

∂σvM n+1eq

∂γ=

√

3

2

1

σvM n+1eq

sn+1 :∂sn+1

∂γ

= −√

3

2

2G

σvM n+1eq

sn+1 :(S : nn+1

)(B.6)

The latest equality is valid since (derivation of B.4 with respect to γ):

∂P : sn+1

∂γ= 0

=∂P

∂γ: sn+1 + P :

∂sn+1

∂γ

= 2Gn+ P :∂sn+1

∂γ

Thus

P :∂sn+1

∂γ= −2Gn (B.7)

or,∂sn+1

∂γ= −2GS : n (B.8)

for which, once more, the latest equality is only a notation, a linear system has

to be solved to compute∂sn+1

∂γ.

The last assumption to solve the plastic problem is to use only an isotropic

hardening law for the yield limit:

σy(εp,d) = σ0

y + hεp,d thus σn+1y = σn

y + h(εp,dn+1 − εp,dn )

43

with h the hardening parameter: h =dσydεp,d

. We thus have

∂σn+1y

∂γ= h

∂(εp,dn+1 − εp,dn )

∂γ(Equ. 26 and B.8)

=

√

2

3h

(

||sn+1|| − 2Gγ

||sn+1||sn+1 :

[S : nn+1

])

(B.9)

Finally, the update of the plastic multiplier is given by combining B.3, B.6, and

B.9:

∆γ =||sn+1|| −

√

2

3σy(γi, s

n+1)

2G

σn+1eq

sn+1 : [S(γi) : nn+1] +2

3h

(

||sn+1|| − 2Gγi

||sn+1||sn+1 : [S(γi) : nn+1]

)

Once more, we should insist on the fact that S is used only as a notation. At

each iteration of the Newton-Raphson procedure used to compute the plastic

multiplier, two linear systems have to be solved to get sn+1 (B.4) and∂sn+1

∂γ(B.7).

Appendix C. Computation of a Consistent Material Tangent Stiff-

ness Operator

This appendix relates the details of the equations needed to derive the consistent

tangent operator for the proposed integration procedure.

The material tangent stiffness operator can be written :

H =dσ

dEN= H

vol. +Hdev. = I ⊗ dp

dEN+

ds

dEN(C.1)

with Hvol. and Hdev. the volumic and deviatoric parts of the tangent stiffness

operator and σ = M−1 : σ and σ = σ(EN ,H , γ):

Hvol. = I ⊗(

p∂(1− ηdH)

∂H:dH

dEN+ (1 − ηdH)

∂p

∂EN

)

Hdev. =∂s

∂EN+

∂s

∂H:dH

dEN+∂s

∂γ⊗ dγ

dH

(C.2)

44

with H = H0 + g, p = p(J) with J = tr(

EN)

and s = s(EN, γ,H)

Hvol. = I ⊗

(

−η3p∂tr (d)

∂H:dH

dEN+ (1− ηdH)

∂p

∂EN

)

Hdev. = M−1 :∂s

∂EN+

(∂M−1

∂H: s+M−1 :

∂s

∂H

)

:dH

dEN+M−1 :

∂s

∂γ⊗ dγ

dEN

(C.3)

Stress tensor partial derivative

One gets, almost trivially for the partial derivatives with respect to EN (see

appendix Appendix B for the details on the derivative with respect to γ - the

derivative with respect to H is computed in the same way.):

∂p

∂EN=∂p

∂J

∂J

∂EN= K

∂J

∂EN(C.4)

∂s

∂EN=

∂s

∂EN

:∂E

N

∂EN(C.5)

= 2GS :∂E

N

∂EN(C.6)

with

∂J

∂EN= I

∂EN

∂EN= 1 (C.7)

Computation ofdγ

dEN(isotropic hardening only)

The derivation ofdγ

dENuses the stationarity property of the yield function:

df

dEN= 0.

df

dEN= 0 (C.8)

=dσvM

eq (s)

dEN− dσy(s, γ)

dEN(C.9)

=

(

∂σvMeq

∂s− ∂σy

∂s

)

:ds

dEN− ∂σy

∂γ

dγ

dEN(C.10)

45

with :

∂σvMeq

∂s=

√

3

2

s

||s|| (C.11)

∂σy∂s

=

√

2

3hγ

s

||s|| (C.12)

∂σy∂γ

=

√

2

3h||s|| (C.13)

ds

dEN=

∂s

∂EN+

∂s

∂H:dH

dEN+∂s

∂γ⊗ dγ

dEN(C.14)

One therefore gets:(√

3

2−√

2

3hγ

)

s

||s|| :ds

dEN−√

2

3h||s|| dγ

dEN= 0

which leads to:√

3

2

s

||s|| :(

∂s

∂EN+∂s

∂γ⊗ dγ

dEN+

∂s

∂H:dH

dEN

)

−h√

2

3

(

||s|| dγdEN

+ γs

||s|| :(

∂s

∂EN+∂s

∂γ⊗ dγ

dEN+

∂s

∂H:dH

dEN

))

= 0

(C.15)

The terms of this equation can be re-arranged to obtain an equation fordγ

dEN:

[(√

3

2− h

√

2

3γ

)

s

||s|| :∂s

∂γ− h

√

2

3||s||

]

dγ

dEN=

h

√

2

3γ

s

||s|| :(

∂s

∂EN+

∂s

∂H:dH

dEN

)

−√

3

2

s

||s|| :(

∂s

∂EN+

∂s

∂H:dH

dEN

)

(C.16)

This givesdγ

dEN= α+ ζ :

dH

dEN(C.17)

with

α =1

den

(

A1 :∂s

∂EN

)

(C.18)

ζ =1

den

(

A1 :∂s

∂H

)

(C.19)

46

and

A1 =

(

h

√

2

3γ −

√

3

2

)

s

||s|| (C.20)

den = −A1 :∂s

∂γ− h

√

2

3||s|| (C.21)

Computation ofdH

dEN

The derivative of H with respect to the natural strain (EN ) depends explicitly

on the damage evolution law (terms written as∂g

∂.):

dH

dEN=

dg

dEN=

∂g

∂EN+∂g

∂s:ds

dEN+∂g

∂p⊗ dp

dEN+

∂g

∂H:dH

dEN+∂g

∂γ⊗ dγ

dEN

(C.22)

Using C.2 and C.17 allows one to write C.22 as:

[

I⊗I − ∂g

∂s:

(∂s

∂γ⊗ ζ +

∂s

∂H

)

− ∂g

∂p⊗ ∂p

∂H− ∂g

∂H− ∂g

∂γ⊗ ζ

]

:dH

dEN=

∂g

∂EN+∂g

∂s:

(∂s

∂EN+∂s

∂γ⊗α

)

+∂g

∂p⊗ dp

dEN+∂g

∂γ⊗α

(C.23)

where all the derivatives both on the left and right hand side of the equation

are known.

This equation is generically written AijklXklmn = Bijmn where A and B are

known. This system solution can be broken down solving 9 equations of the

type Aijklxkl = bij which are systems of 9 linear equations in xkl.

Derivatives of M and M−1

Derivatives of M and M−1 with respect to damage remain to be computed.

These are in general sixth order tensors. However, in the computation of the

stiffness operator, the general sixth order tensors are not needed. Only the

fourth order tensors∂M

∂H: s,

∂M−1

∂H: s as well as

∂tr (d)

∂Hare to be computed.

47

Computation of∂M

∂H: s

[M : s]ij = [dev (HsH)]ij = [HsH ]ij −1

3[HsH ]nn δij

Therefore

∂ [M : s]ij∂Hkl

∣∣∣∣s cst

= δik [Hs]jl + δjl [Hs]ik − 1

3([Hs]kl + [Hs]lk) δij (C.24)

∂M

∂H: s = I⊗(Hs) + (Hs)⊗I − 1

3I ⊗

[(Hs) + (Hs)T

](C.25)

Computation of∂M−1

∂H: s

[M

−1 : s]

ij= (H−1)iosop(H

−1)pj︸ ︷︷ ︸

⋆ij

−H−2

ij (smnH−2mn)

tr(H−2

)

︸ ︷︷ ︸

⋆⋆ij

(C.26)

One has

∂⋆ij∂Hkl

∣∣∣∣s cst

=∂H−1

io

∂HklsopH

−1pj +H−1

io sop∂H−1

pj

∂Hkl

= −H−1ik

(H−1sH−1

)

jl−(H−1sH−1

)

ikH−1

jl

and

∂ ⋆ ⋆ij∂Hkl

∣∣∣∣s cst

=∂H−2

ij

∂Hkl

smnH−2mn

tr(H−2

)+H−2ij

smn

tr(H−2

)∂H−2

mn

∂Hkl−H−2

ij smnH−2mn

(tr(H−2

))2∂tr

(H−2

)

∂Hkl

with

∂H−2ij

∂Hkl=∂H−1

io

∂HklH−1

oj +H−1io

∂H−1oj

∂Hkl= −(H−1

ik H−2jl +H−2

ik H−1jl )

∂H−2

∂H= −(H−2⊗H−1 +H−1⊗H−2)

and∂tr

(H−2

)

∂H=∂H−2 : I

∂H= −2H−3

48

One can therefore write

∂ ⋆ ⋆ij∂Hkl

∣∣∣∣s cst

= −[(

H−1ik H

−2jl +H−2

ik H−1jl

) smnH−2mn

tr(H−2

)

+H−2

ij

tr(H−2

)([H−1sH−2]kl + [H−2sH−1]kl

)−H−2

ij

2smnH−2mn

(tr(H−2

))2H−3

kl

]

Finally the fourth order tensor∂M−1 : s

∂Hcan be written :

∂M−1ijop

∂Hklsop = −H−1

ik

(H−1sH−1

)

jl−(H−1sH−1

)

ikH−1

jl

+smnH

−2mn

tr(H−2

)

[

H−1ik H

−2jl +H−2

ik H−1jl − 2

tr(H−2

)H−2ij H

−3kl

]

+H−2

ij

tr(H−2

)([H−1sH−2]kl + [H−2sH−1]kl

)(C.27)

or else

∂M−1

∂H: s = −H−1⊗

(H−1sH−1

)−(H−1sH−1

)⊗H−1

+s : H−2

tr(H−2

)

[

H−1⊗H−2 +H−2⊗H−1 − 2

tr(H−2

)H−2 ⊗H−3

]

+H−2

tr(H−2

) ⊗[H−1sH−2 +H−2sH−1

](C.28)

Computation of∂tr (d)

∂HAs d = I − H−2 and as, for any invertible symmetric tensor such as H , one

has:∂H−1

∂H= −H−1⊗H−1, so that one can write:

∂d

∂H= H−2⊗H−1 +H−1⊗H−2 (C.29)

The computation of∂tr (d)

∂His therefore directly given by:

∂tr (d)

∂H=

∂d

∂H: I =

(H−2⊗H−1 +H−1⊗H−2

): I = 2H−3 (C.30)

As H is a polynomial function of d, the derivative given by C.29 can be com-

puted by the use of a spectral decomposition of d.

49

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